Infinite-dimensional projective space

Let us remark that the infinite-dimensional projective space RP°° appearing in (8.4) could also be defined as a colimit of the embedding sequence of the finite-dimensional projective spaces... 

In the noise-free case, ) = ) = 0, one can calculate the null isoclines of the system. These are plotted in Fig. 5.19 using the current-voltage projection of the originally infinite-dimensional phase space. There are three... 

Since a complex scalar product resembles the EucUdean dot product in its form and definition, we can use our intuition about perpendicularity in the Euclidean three-space we inhabit to study complex scalar product spaces. However, we must be aware of two important differences. Eirst, we are dealing with complex scalars rather than real scalars. Second, we are often dealing with infinite-dimensional spaces. It is easy to underestimate the trouble that infinite dimensions can cause. If this section seems unduly technical (especially the introduction to orthogonal projections), it is because we are careful to avoid the infinite-dimensional traps. 

We know how the extension of affine tangent systems to projective spaces can be achieved geometrically. An imaginary or virtual point is added to each bundle of parallel lines. This virtual point serves as the point "infinitely far" for each bundle of lines. The points at infinity are called collinear if and only if they are the infinitely remote points of three lines in the same plane. Four points at infinity are known as coplanar if they are the infinitely remote points of four lines in the same three-dimensional space. At the same time it should be emphasized that this introduction of points at infinity was done in each tangent space. 

Between two such peaks is a sequence of further peaks of lower intensity that all lie in an infinite set of coincident interpenetrating Fibonacci sequences of arbitrarj origin. (They can be labelled through a projection from Euclidean two-dimensional space as described in the main text.)... 

The profile likelihood function of 9i is shown in three-dimensional space in Figure 1.9. The two-dimensional profile likelihood function is found by projecting it back to the f X 9i plane and is shown in Figure 1.10. (It is like the "shadow" the curve L 9i, 02 9i, data) would project on the f x6i plane from a light source infinitely far away in the 02 direction.) The profile likelihood function may lose some information about 9 compared to the joint likelihood function. Note that the maximum profile likelihood value of 9 will be the same as its maximum likelihood value. However, confidence intervals based on profile likelihood may not be the same as those based on the joint likelihood. 

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